Distinct and Complete Integer Partitions George Beck, Wolfram - PDF document

Distinct and Complete Integer Partitions George Beck, Wolfram Research This is joint work with George Andrews and Brian Hopkins. Abstract Two infinite lower-triangular matrices related to integer partitions are inverses of each other. One matrix comes from an analogue of the Möbius mu function, while the other comes from counting generalized complete partitions; a complete partition of n has all possible subsums 1 to n. Mathematica Definitions Integer Partitions Definition A multiset is a collection of elements (like a set) where an element can occur a finite number of times (unlike a set). An integer partition λ of a positive integer n is an multiset of positive integers λ i (called its parts) that sum to n . We write λ = ( λ 1 , λ 2 , … λ m ) ⊢ n . Mathematically we use (round) parentheses and in Mathematica we use (curly) braces, which denotes an (ordered) list, not a set. For example, ( 3, 1, 1 ) ⊢ 5. Since the elements of a multiset are unordered (like a set), we can take them to be in nonincreasing order from now on. Here are the integer partitions of 5: ����  �� {{ 5 } , { 4, 1 } , { 3, 2 } , { 3, 1, 1 } , { 2, 2, 1 } , { 2, 1, 1, 1 } , { 1, 1, 1, 1, 1 }} Here they are again more compactly: ����  �� { 5, 41, 32, 311, 221, 2111, 11111 } Other Definitions An older alternative definition is along these lines: “A partition is a way of writing an integer n as a sum of positive integers where the order of the addends

2 ��� Distinct and Complete Integer Partitions.nb is not significant, … . By convention, partitions are normally written from largest to smallest addends … , for example, 10 = 3 + 2 + 2 + 2 + 1.” (mathworld.wolfram.com/Partition.html) With such a definition, 3 + 2 + 2 + 2 + 1 has to be frozen, because as an arithmetic expression it is 10 and the parts are gone. Yet another definition: λ = ( λ 1 , λ 2 , λ 3 , … , λ m ) is an partition of n if the finite sequence λ = ( λ 1 , λ 2 , … , λ m ) is such that λ 1 ≥ λ 2 ≥ … ≥ λ m and λ 1 + λ 2 + … + λ m = n . Ferrers Diagram For each part λ i of a partition λ , draw a row of λ i dots, then stack the rows. ● ● ����  �� ● ● Conjugate Partition The conjugate partition λ ' of a partition λ is the partition corresponding to the transpose of the Ferrers diagram of λ . ● ● ● ����  �� ● So ( 3, 1 ) is the conjugate partition of ( 2, 1, 1 ) and vice versa. Distinct Partition A distinct partition has no repeated part. Here are the four distinct partitions of 6. ����  �� { 6, 51, 42, 321 } The remaining partitions of 6 have repeated parts. ����  �� { 33, 222, 411, 2211, 3111, 21111, 111111 } This is the sequence counting the number of distinct partitions of n . ����  �� { 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64 } Generating Functions Number of Partitions The number of partitions of n is 1, 2, 3, 5, 7, 11, … but the next number is not 13: ����  �� { 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77 } The generating function for this sequence p ( n ) is:

Distinct and Complete Integer Partitions.nb ��� 3 ����  �� x + 2 x 2 + 3 x 3 + 5 x 4 + 7 x 5 + 11 x 6 + 15 x 7 + 22 x 8 + 30 x 9 + 42 x 10 + 56 x 11 + 77 x 12 + … 1 The generating function is equal to the infinite product ∏ 1 - x i : ∞ i = 1 Number of Distinct Partitions The number of distinct partitions of n : ����  �� { 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15 } The generating function for this sequence q ( n ) is: ����  �� x + x 2 + 2 x 3 + 2 x 4 + 3 x 5 + 4 x 6 + 5 x 7 + 6 x 8 + 8 x 9 + 10 x 10 + 12 x 11 + 15 x 12 + … ∞  1 + x i  : It is equal to the infinite product ∏ i = 1 Two Möbius Functions Square-Free Numbers A square-free integer is one that is not divisible by a square greater than 1. Here are the square-free numbers up to 100: ����  �� { 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97 } Here are numbers up to 100 that are not square-free: ����  �� { 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100 } Möbius Function μ In multiplicative number theory the Möbius μ function is defined on the positive integers as follows. 1. If n is not square-free, μ ( n ) = 0. 2. If n is square-free, then n can be written as the product of m distinct primes, for some positive integer m . In that case, μ ( n ) = (- 1 ) m . In other words, μ of a square-free integer is - 1 or 1 according to whether n has an odd or an even number of prime factors. For example, μ ( 4 ) = 0, μ ( 5 ) = - 1, μ ( 6 ) = 1. Möbius Partition Function μ P The function μ P is the partition analogue of the ordinary Möbius function μ .

4 ��� Distinct and Complete Integer Partitions.nb μ μ P product partition ����  �� primes factors parts square-free distinct Definition of μ P : 1. Let μ P ( λ ) = 0 if the partition λ has a repeated part. 2. If the partition λ has distinct parts and m parts in all, μ P ( λ ) = (- 1 ) m . Here are the partitions of 6 and the corresponding values of the Möbius partition function μ P : 6 - 1 51 1 42 1 411 0 33 0 321 - 1 ����  �� 3111 0 222 0 2211 0 21111 0 111111 0 Infinite Triangular Matrices Pascal’s Triangle The prime example of an infinite lower-triangular matrix is Pascal’s triangle T . Imagine that the rows keep going down and the columns keep going to the right. For readability, replace zeros with dots. ����  �������������� 1 · · · · · · · · · 1 1 · · · · · · · · 1 2 1 · · · · · · · 1 3 3 1 · · · · · · 1 4 6 4 1 · · · · · 1 5 10 10 5 1 · · · · 1 6 15 20 15 6 1 · · · 1 7 21 35 35 21 7 1 · · 1 8 28 56 70 56 28 8 1 · 1 9 36 84 126 126 84 36 9 1 Here is the matrix product T · T .

Distinct and Complete Integer Partitions.nb ��� 5 ����  �������������� 1 · · · · · · · · · 2 1 · · · · · · · · 4 4 1 · · · · · · · 8 12 6 1 · · · · · · 16 32 24 8 1 · · · · · 32 80 80 40 10 1 · · · · 64 192 240 160 60 12 1 · · · 128 448 672 560 280 84 14 1 · · 256 1024 1792 1792 1120 448 112 16 1 · 512 2304 4608 5376 4032 2016 672 144 18 1 Here is the matrix inverse of T . ����  �������������� 1 · · · · · · · · · - 1 1 · · · · · · · · 1 - 2 1 · · · · · · · - 1 3 - 3 1 · · · · · · 1 - 4 6 - 4 1 · · · · · - 1 5 - 10 10 - 5 1 · · · · 1 - 6 15 - 20 15 - 6 1 · · · - 1 7 - 21 35 - 35 21 - 7 1 · · 1 - 8 28 - 56 70 - 56 28 - 8 1 · - 1 9 - 36 84 - 126 126 - 84 36 - 9 1 Stirling Numbers of the First and Second Kind The Stirling numbers of the first and second kind are another example of a pair of inverse lower-triangu- lar matrices. A Stirling number of the first kind counts how many permutations of { 1, 2, … , n } have k cycles. ����  �������������� 1 · · · · · · · - 1 1 · · · · · · 2 - 3 1 · · · · · - 6 11 - 6 1 · · · · 24 - 50 35 - 10 1 · · · - 120 274 - 225 85 - 15 1 · · 720 - 1764 1624 - 735 175 - 21 1 · - 5040 13068 - 13132 6769 - 1960 322 - 28 1 A set partition of a finite set, say T = { 1, 2, 3, … , n } , is a set of disjoint nonempty subsets of T . A Stirling number of the second kind counts how many set partitions of { 1, 2, … , n } have k subsets. ����  �������������� 1 · · · · · · · 1 1 · · · · · · 1 3 1 · · · · · 1 7 6 1 · · · · 1 15 25 10 1 · · · 1 31 90 65 15 1 · · 1 63 301 350 140 21 1 · 1 127 966 1701 1050 266 28 1 The two matrices are inverses of each other.